Petruccelli, Joseph D.; Woolford, Samuel W. A threshold AR(1) model. (English) Zbl 0541.62073 J. Appl. Probab. 21, 270-286 (1984). Let \(\{a_ t\}\) be i.i.d. random variables with mean 0. Consider the model \(Z_ t=\Phi_ 1Z^+_{t-1}+\Phi_ 2Z^-_{t-1}+a_ t\), where \(\Phi_ 1\) and \(\Phi_ 2\) are real parameters and \(x^+=\max(x,0)\), \(x^-=\min(x,0)\). It is proved that the process \(\{Z_ t,t\geq 0\}\) is ergodic iff \(\Phi_ 1<1\), \(\Phi_ 2<1\) and \(\Phi_ 1\Phi_ 2<1\). (This is very surprising, since one would expect the condition \(| \Phi_ 1|<1\), \(| \Phi_ 2|<1.)\) Further, it is shown that the least squares estimators of \(\Phi_ 1\) and \(\Phi_ 2\) are consistent and asymptotically normal, and a test of the hypothesis \(\Phi_ 1=\Phi_ 2\) is proposed. The asymptotic formulas are complemented by small-sample results from simulated data. Reviewer: J.Anděl Cited in 2 ReviewsCited in 60 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60J05 Discrete-time Markov processes on general state spaces Keywords:non-linear time series; TAR models; autoregressive models; Markov chains; stationary distribution; ergodic process; least squares estimators; small-sample results PDFBibTeX XMLCite \textit{J. D. Petruccelli} and \textit{S. W. Woolford}, J. Appl. Probab. 21, 270--286 (1984; Zbl 0541.62073) Full Text: DOI Link